On the Lattice of L-closure Operators

In 1965 L. A Zadeh [11] introduced fuzzy sets as a generalization of ordinary sets. After that C. L. Chang [2] introduced fuzzy topology and that led to the discussion of various aspects of L-topology by many authors. The Čech closure spaces introduced by Čech E. [1] is a generalization of the topological spaces. The theory of fuzzy closure spaces has been established by Mashhour and Ghanim [4] and Srivastava et. al [6],[7]. The definition of Mashhour and Ghanim is an analogue of Čech closure spaces and Srivastava et. al. have introduced it as an analogue of Birkhoff closure spaces in [7]. Based on [7], Rekha Srivastava and Manjari Srivastava studied the subspace of a fuzzy closure space. The notion of T0-fuzzy closure spaces and T1 fuzzy closure spaces were also introduced in [6]. In [5] P. T. Ramachandran studied the properties of lattice of closure operators. In [3] T. P. Johnson studied some properties of the lattice L(X ) of all fuzzy closure operators on a fixed set X . In [9] Wu-Neng Zhou introduced the concept of L-closure spaces and the convergence in L-closure spaces. In this paper we study the lattice LC (X ) of Lclosure operators and L-closure spaces which is a generalization of the concept of fuzzy closure spaces. Here we proved that the complete lattice LC (X ) is not modular. Also we identify the infra L-closure operator and ultra L-closure operator and establish the relation between ultra L-topology and ultra L-closure operator. We proved that an L-closure operator is an ultra Lclosure operator if and only if it is the L-closure operator associated with an ultra L-topology. Also proved that infra Lclosure operators are less than or equal to any nonprincipal ultra L-closure operator and no nonprincipal ultra L-closure operator has a complement so that the lattice of L-closure operators is not complemented in general.